The likelihood function defines the probability of the observed trial-by-trial data as a function of the parameters in a model. In the PTM (
Figures 2a, c, e), signal discriminability
d′ in condition (
c,
Next,
cue), where
c is the pseudo-character signal contrast,
Next is the standard deviation of external noise contrast, and
cue is either pre or simultaneous, is a function of six model parameters (
Na,
Nm, β, γ,
Aa, and
Af) and two stimulus parameters (
c and
Next;
Dosher & Lu, 2000b;
Lu & Dosher, 1998;
Lu & Dosher, 2000):
\begin{eqnarray}
&&d^{\prime}(c,{N_{ext}},cue, N_{a},{N_m},\beta ,\gamma ,{A_a},{A_f}) = \nonumber \\
&&\frac{{{{\left( {\beta c} \right)}^\gamma }}}{{\sqrt {{{\left( {{A_f}{N_{ext}}} \right)}^{2\gamma }} + N_m^2\left[ {{{\left( {\beta c} \right)}^{2\gamma }} + {{\left( {{A_f}{N_{ext}}} \right)}^{2\gamma }}} \right] + {{\left( {{A_a}{N_a}} \right)}^2}} }},\!\!\!\!\! \nonumber\\
\end{eqnarray}
where
Na is the standard deviation of the internal additive noise,
Nm is the proportional constant of the multiplicative noise, β is the gain of the perceptual template, γ is the exponent of the transducer,
Aa reflects internal additive noise reduction by attention,
Af reflects external noise exclusion by attention.
1 Aa and
Af depend on the cuing condition. The probability of obtaining a correct response in a single trial is:
\begin{eqnarray}
&& {p_{correct}}(c,{N_{ext}},cue,{N_a},\;{N_m},\beta ,\gamma ,{A_a},{A_f}) \nonumber \\
&&\quad =\mathop \int _{ - \infty }^{ + \infty } g ( {x - d^{\prime}(c,{N_{ext}},cue,\;{N_a},\;{N_m},\beta ,\gamma ,{A_a},{A_f}} ))\!\!\!\!\! \nonumber \\
&&\quad\times\;{G^3}\left( x \right)dx,
\end{eqnarray}
where g(.) and
G(.) are the probability density and cumulative probability functions of a standard Gaussian distribution. The probability of obtaining
M correct responses from a total of
T trials in a single condition is described by a binomial distribution
B:
\begin{eqnarray}
&&p(M|c,{N_{ext}},cue,T,{N_a},{N_m},\beta ,\gamma ,{A_a},{A_f}) \nonumber \\
&&=B ( {{p_{correct}}(c,{N_{ext}},cue,{N_m},\beta ,\gamma ,{A_a},{A_f}} ),M,T).\;\quad
\end{eqnarray}
Figure 2 illustrates the signature performance effects of several attention mechanisms on the TvC functions, which graph contrast threshold as a function of the contrast of external noise. These functions are useful in illustrating the consequences of attention mechanisms. Specifically,
stimulus enhancement reduces contrast thresholds in the region of zero or low external noise (see
Figures 2a, b), accounting for effects of attention in the absence of external noise. Mathematically equivalent to internal additive noise reduction, it corresponds to claims of perceptual enhancement (
Posner, Nissen, & Ogden, 1978).
External noise exclusion reduces contrast thresholds in the region of high external noise (see
Figures 2c, d), where there is external noise to exclude, by focusing perceptual analysis on the appropriate time, spatial region, and/or content characteristics of the signal stimulus (
Dosher & Lu, 2000b;
Shiu & Pashler, 1994).
Multiplicative internal noise reduction reduces contrast thresholds throughout the entire range of external noise levels (see
Figures 2e, f). In addition, measuring TvC functions at two or more criterion performance levels along the psychometric function resolves the individual contribution of each mechanism in when multiple mechanisms are involved (
Dosher & Lu, 2000a;
Lu & Dosher, 2000). In prior applications of the PTM, only stimulus enhancement and external noise exclusion have been observed, so these two mechanisms of attention are examined in our analysis.
In what follows, we use the notation
θij to denote the PTM parameters for individual
i in the
jth test, which is the
jth repetition of the whole experiment with all the stimulus contrast, external noise, and cuing conditions.
Sijk,
Tijk, and
Mijk denote, respectively, the stimulus parameters (
c,
Next, and cue), the numbers of total trials and correct responses for individual
i in the
kth condition of the
jth repetition, where condition
k denotes each combination of
c,
Next, and cue.
Table 1 shows a model structure with external noise exclusion in central cuing and a combination of external noise exclusion and internal noise reduction in peripheral cuing, which agrees with the previous analysis of the study (
Lu & Dosher, 2000). Later, we will consider additional models in which central cuing causes both external noise exclusion and internal noise reduction, and a model in which attention has no effect. The system nonlinearity parameter γ of the PTM is assumed to be equal in central and peripheral cuing, consistent with many applications of the PTM (
Dosher & Lu, 2000a;
Dosher & Lu, 2000b;
Lu & Dosher, 2000). (By convention, the
Aa and
Af parameters are set to 1 in simultaneous, or unattended, conditions, and in both conditions if the respective attention mechanisms are not effective.
Aa and
Af < 1 if attention improves performance.)
We express the probability of obtaining
Mijk correct responses in
Tijk trials as:
\begin{eqnarray}
&& p(M_{ijk}|{\bf{S}}_{ijk},{T_{ijk}},{{\boldsymbol\theta} _{ij}}) =
B(p_{correct}({\bf{S}}_{ijk},\;{{{\boldsymbol\theta} _{ij}}} ),{M_{ijk}},{T_{ijk}}).\!\!\!\!\!\!\nonumber\\
\end{eqnarray}
The likelihood of obtaining the entire dataset for a given set of parameters
θ1: I, 1: J, is:
\begin{eqnarray}
&& p({{\boldsymbol M}_{1:I,1:J,1:K}}|{{\bf{S}}_{1:I,1:J,1:K}},{{\boldsymbol T}_{1:I,1:J,1:K}},{{\boldsymbol \theta} _{1:I,1:J}}) \nonumber \\
&&\qquad = \mathop \prod \limits_{i = 1}^I \mathop \prod \limits_{j = 1}^J \mathop \prod \limits_{k = 1}^{{K_i}} p({M_{ijk}}|{{\bf{S}}_{ijk}},{T_{ijk}},{{\boldsymbol \theta} _{ij}}).\quad
\end{eqnarray}
In this study, we set J = 1 because all the individuals only repeated the experiment once and we want to estimate the PTM parameters from the whole experiment, k runs through all the (c, Next, and cuing) combinations for each individual, with Ki = 288 for the three observers who participated in both the central and peripheral cuing experiments, and Ki = 144 for the two observers who only participated in the central cuing experiment.